John,
I'm glad you asked about infinity because it shows why combinatorics can give a conceptual understanding of quantum. The typical infinity thread leads to a mention of fractals. Fractals require the same type of structures on all scales. If the universe is anything close to being a fractal I would expect the problem of gravity in quantum mechanics to be much easier. Unfortunately, the fundamental quantum mechanical nature of the universe is very dissimilar to its large scale structures.
I'd like to summarize my above argument on infinity because I think it is a very important topic:
Quantum mechanics and experiments give finite real numbers:
[-n...-4,-3,-2,-1,0,1,2,3,4...n]
How should physicists and mathematicians calculate the finite real numbers to give valid predictions?
With:
1) Infinite real numbers?
(...-2,-1,0,1,2...)
2) What about finite complex (imaginary) numbers?
[-ni...-2i,-1i,0,1i,2i...ni]
3) Infinite complex numbers?
(...-2i,-1i,0,1i,2i...)
The best, with minimum assumptions, is to calculate the finite real numbers using other finite real sets. It is impossible to prove infinity via experiment. The proof in mathematics usually comes from induction:
1) Basis: Does it work for n = 0?
2) Inductive step: If it works for n does it work for n+1?
A proof by induction works for the infinite or the finite, but it does not tell us if our universe is infinite.
The quanta in quantum mechanics means discrete and that these numbers are not arbitrarily small. Here is a simplified argument with mathematical hyperbole:
E = nh ---> [E1 = h, E2 = 2h, E3 = 3h ----> [1,2,3,...?
It is incorrect to have an energy that can be E= 1.34645h Energy is a quanta, whole numbers, integers.
I think we should focus on the real numbers; pure mathematics can not tell us if our universe is finite or infinite. An experiment would only tell us a finite n value. If a later experiment finds n+1 then we still do not know if there is an n+2. George and I are playing it safe. We think an argument that says, "infinity explains everything" is not experimentally nor mathematically rigorous. An idea using finite real sets can be confirmed and falsified via experiment.
How many angels can fit on a pin head?
Prove it.
Sorry for the ranting,
Brian